A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

learn more… | top users | synonyms

4
votes
1answer
28 views

How to put my knowledge of probability and statistics to practice

Background: I am a masters student in stochastic analysis. My course is very theoretical, which in general is fine by me, it is what I enjoy the most. From the more data-friendly subjects, I have (or ...
0
votes
1answer
20 views

Moments of a random sum with bounds Poisson distributed?

We have that $N$ and ${X_1,X_2,\dots}$ are all independent and that $f(x)=Cx^2(1-x)^2$. Then, we have: $$Z=\sum_{j=1}^{N+1}X_j$$ $N$~Poisson$\lambda$. Find the expectation and the variance of $Z$. ...
1
vote
0answers
38 views

Kolmogorov continuity theorem on Wikipedia.

I am wondering if the Kolmogorov continuity theorem on Wikipedia is wrong? : They say that the modification is sample continuous, and when we click on that link it says that it is a.s. continuous. ...
1
vote
1answer
19 views

Example of a non square-integrable martingale?

Are there (simple) examples of martingales which aren't square integrable?
0
votes
0answers
11 views

What are some modern books on Markov Chains with plenty of good exercises?

I would like to know what books people currently like in Markov Chains (with syllabus comprising discrete MC, stationary distributions, etc.), that contain many good exercises. Some such book on ...
1
vote
4answers
61 views

$2$ players take turns and draw from a box containing $1000$ balls, $3$ of them are black.

I'm not sure how to tackle this question. Assume a box containing $1000$ balls, $3$ of them are black and the rest are white. $2$ players $A_1$ & $A_2$ take turns and draw from the box without ...
1
vote
1answer
45 views

If $dX_{t} = X_{t}\,dt + \,dB_{t}$, why does $e^{- t}dX_{t} = e^{-t} X_{t} \,dt + e^{-t} \,dB_{t}$?

I'm taking a course in stochastic differential equations, and in order to solve $dX_{t} = X_{t}\,dt + \,dB_{t}$, the book gives a hint: to multiply both sides of this equation by $e^{-t}$. (But, as ...
3
votes
1answer
34 views

Which inequalities are there with stochastic integration?

Which inequalities can I use with stochastic integration? For example, with the standard lebesgue integral we have $$\left|\int_\Omega f(x) dx\right| \le M |\Omega|$$ (where $M$ is the maximum of ...
0
votes
1answer
33 views

How to show that this is a martingale?

Let $H_s$ be a predictable and bounded process. How can I show that $$M_t = \int_0^t H_s \, dW_s$$ is a martingale? Clearly since $H_s \in L^2_\text{loc} (W)$ we have that $M_t$ is a local ...
1
vote
0answers
22 views

The “how many pieces do you have buy on average” problem, a markov problem?

I recently discovered a problem similar to this one in a book about Markov chains: Assume you can buy $n-$ different set of cards in a store, but you do not know which one you'll buy: What is the ...
0
votes
0answers
22 views

Are there different definitions of a continuous time Markov chain, condition on a finite or infinite number of earlier values?

My book defines a continuous time Markov chain like this. Let $\{X_t\}, t \in T$ be a stochastic process on $(\Omega, \mathcal{A}, P)$, with a countable state space $S$. The process is a Markov ...
1
vote
0answers
25 views

Poisson Process with continuous rate, Finding Conditional Number of Arrivals

Poisson with customer arrival to the shop rate given by $\lambda (t)=16-(t-4)^2$ Calculate $P(N(5)-N(3)=40|N(4)=70)$ where $N(i)$ means the number of arrivals in the first $i$ hours. The shop ...
3
votes
0answers
28 views

Markov Chain: Steady State Distribution.

A total of $M$ balls are divided between two urns A and B. A ball is chosen uniformly at random. If it is chosen from urn A then it is placed in urn B with probability $b$ and otherwise it is returned ...
0
votes
1answer
18 views

What is the difference between a martingale and doob's martingale?

Every sequence that was termed as a doob's martingale, I was able to deduce that it was also a martingale. So here are few of my questions: 1) Is it correct to say that every doob martingale is also ...
0
votes
1answer
21 views

Prove a conditional distribution is uniformly distributed across a given interval?

$X$ and $Y$ are independent random variables identically exponentially distributed with $\lambda$. Take $Z=X+Y$. Show that $(X|Z=z)$ is uniformly distributed over $(0<x<z)$. Then, find ...
1
vote
1answer
15 views

Doubt with Notation on Conditional Expected Value Demonstration

I´m having trouble writing a demonstration for the Conditional Expected Value using $\sigma$-algebra. I know its really simple and actually logic but I just can´t find the way to write it. Hope anyone ...
0
votes
0answers
16 views

Continuity of random variable as function of a random variable

Suppose, we are given a continuos random variable $X$ and a continuous and nondecreasing function $f$. Can it be shown that a second random variable $Y=f(X)$ is continuos on the support of $X$? What ...
0
votes
1answer
28 views

Marginal distribution from a Poisson distribution where intensity is exponentially distributed?

Given that $N$ is Poisson distributed with a random intensity $Y$, the conditional distribution of $(X|Y)$ is defined as, for $n=0,1,\dots$ $$P[N=n|Y=\lambda]=e^{-\lambda}\lambda^n\frac{1}{n!}$$ $Y$ ...
0
votes
1answer
28 views

Poisson Probability with rate $\lambda (t)=-(t-4)^2+16$

The rate at which customer arrive to the bookstore is $\lambda (t)=-(t-4)^2+16 $ where $t$ measured in hours. The customers can buy a book with probability $0.5$ and they can also buy a coffee ...
2
votes
1answer
33 views

Probability matrices in an online game or how to approach matching players to maps to achieve better user experience

Probably I had nonstandart question, but I hope to find some help and valueable advice. Assume I have an online game with $n$ players (let's say $n$ is about 100.000). There's also $m$ maps ($m$ is ...
0
votes
1answer
20 views

Random sum of normal distribution with bounds Poisson distributed?

A random variable, $M$, is Poisson distributed with $\lambda=2$. ${X_1,X_2,\dots}$ are independently identically distributed random variables with $\mu=3$ and $\sigma=.2$. Introduce a new random ...
0
votes
1answer
22 views

Stochastic process independent of its future

Are there examples of predictable stochastic processes $X$ such that their past is independent of their future? More formally, such that $\sigma\{X_s | s\in (0,t]\}$ is independent of $\sigma\{X_s | ...
2
votes
0answers
14 views

Ergodicity of the natural measure implies uniqueness of the invariant density?

Consider a dynamical system $x_{t+1} = F(x_t)$ defined in $\Omega$ and its natural measure to be $\mu$. The Perron-Frobenius operator $F$ maps the density $f$ in time according to $f_{t+1} = F(f_t)$. ...
0
votes
0answers
26 views

Ito Formula for Stochastic Integral

Suppose I have $$dS_t = \mu(S_t,t) dt + \sigma(S_t,t)dW_t$$ What would be the process satisfying the following process of $y_t$? $$y_t = \int_0^t S_u du + \int_0^t S_u dW_u$$ I'm not quite sure ...
2
votes
1answer
13 views

How to prove that the following process is a Martingale using Ito's formula?

I am asked to prove that $Y_t$ is a martingale where $Y_t=\exp\left(\int_0^tf(s)\,dW_s-1/2\int_0^tf(s)^2\,dt\right)$ using Ito's formula. After applying Ito's formula (I hope I made no mistake) I get ...
1
vote
2answers
25 views

If $M_t$ is a martingale, is this process a martingale too?

If $M_t$ is a martingale, is this process a martingale too ? $X_t=\int_0^tY_sdM_s$ where $Y_t$ is some process that makes $\int_0^tY_sdM_s$ defined If not, what about the case $Y_t=M_t$ ?
0
votes
0answers
27 views

Explicit formula for return probability of simple random walk

Is there an explicit formula for the probability that a simple symmetric random walk on $\mathbb{Z}$ starting from $1$ will not hit $0$ before time $t$?
2
votes
0answers
23 views

Return time for two independent one dimensional random walks

Let $X^1$ and $X^{-1}$ be two simple random walk in $\mathbb{Z}$ starting respectively from $1$ and $-1$. Let $\tau$ be the first time one of them reaches the origin, $$\tau = \inf \{ j \geq 0 \, : ...
2
votes
0answers
52 views

Proof of the existence of a reversible stationary distribution

$p$ is a finite Markov chain where $p(i,j)>0$ for all $i,j$. Prove a reversible stationary distribution exists for $p$ if $p(i,j)p(j,k)p(k,i)=p(i,k)p(k,j)p(j,i)$ for all $i,j,k$ This question is ...
0
votes
0answers
14 views

Convergence of stochastic processes via convergence of infinitesimal generators

Given a sequence of sequence processes $(X_N(\cdot))_{N \geq 0}$, I want to show this sequence converges to another process $X(\cdot)$ by considering that the sequence of generators $(A_N)_{N \geq 0}$ ...
4
votes
1answer
43 views

A counterexample for supremum of stopping times

Let $\mathbb{F} = \{ \mathcal{F}_t \}_{t \geq 0}$ be a continuous time filteration. $\tau : \Omega \to [0, \infty]$ is called an $\mathbb{F}$-stopping time if $\{ \tau \leq t \} \in \mathcal{F}_t$ for ...
1
vote
0answers
22 views

Distribution of Hitting Times

I am curious about the following problem: Let the diffusion process $\{X_t\}_{t\ge 0}$ be defined as $$dX_t=c(1-X_t)X_td B_t$$ where $X_0\in (a,b)\subseteq (0,1)$, $c>0$, and $B_t$ is the standard ...
0
votes
0answers
15 views

Is the result of a Monte-Carlo simulation of a continuous function and with continuous input distributions again continuous?

Is the result of a Monte-Carlo simulation of a continuos function and with continuos input distributions again continuous? Suppose, we have a continuos function $f$ and a number of continuous random ...
2
votes
2answers
59 views

Modeling with Markov Chains and one-step analysis

I have set up the following model: Let $X_n$ be the number of heads in the $n$-th toss and $P(X_0=0)=1$. I can calculate the transition matrix $P$. Define $$ T=\min\{n\geq 0\mid X_n=5\}. $$ Then ...
0
votes
0answers
26 views

Dealing with Recurrence Relations of Random Variables

Let $(Y_n)_{n\in \mathbb N} $ be some sequence of independent random variables, and $(X_n)_{n\in \mathbb N} $ another sequence, defined recursively as follows: $$X_{n+1} = \alpha X_n + \beta Y_n ...
0
votes
0answers
8 views

How is the form of the transition functions of a process define as the solution of a stochastic differential equation?

I´m searching for a prove that solutions of SDE are markovian and i'm trying to find (or understand) in this case (SDE) what form have the transition functions associated to this kind of process. I´ve ...
2
votes
0answers
30 views

Does the quadratic covariation process define a measure?

In the context of stochastic integration (when we define the space $L^2(M)$), we define the (possibly infinite) measure $$P_M = P \otimes [M]$$ by $$E_M[Y] = E\left[\int_0^\infty Y_s(\omega) ...
0
votes
0answers
14 views

Stationary process vs stationary increments

Am I right that these are not the same, i.e. a stationary process need not have stationary increments and vice versa? example: Brownian motion is not a stationary process but it has stationary ...
2
votes
0answers
11 views

(joint) Functional CLT for partial sums and counting process

Assume you are given a sequence of random variables $(X_i)_{i\geq1}$. Assume moreover that they are sufficiently smooth, say $\mathbb E[X^2]<+\infty$. Define the diffusion-scaled partial sum as ...
1
vote
0answers
28 views

Joint convergence of stochastic processes

Suppose I have processes $X_n(t)\overset{d}{\longrightarrow} X(t)$, and $Y_n(t)\overset{p}{\longrightarrow} ct$ for some constant $c$. Then, can I conclude like in Slutsky's theorem that ...
0
votes
1answer
26 views

A clarification on $L_{loc}^2$ process and stochastic exponential

In the book by A. Pascucci (PDE and Martingale Methods in Option Pricing) I have found the following definition of $\mathbb{L}^2_{\text{loc}}$ process. Later (pp. 329-330) for a process ...
-1
votes
0answers
15 views

Compound Poisson Process - calculate [on hold]

Compound Poisson Process $\{Y_t : 0 \le t \}$ with the intensity $\lambda$ there are breaks with distribution $exp(a)$, a>0. Calculate: $$P \{ Y_t >2 |N_t = 3\}$$. Any ideas? I know that $Y_t$ is ...
1
vote
2answers
54 views

Defining the states when we roll one single die repeatedly

We roll a single die and the game stops as soon as the sum of two successive rolls is either 5 or 7. We want to find the probability that the game stops at a sum of 5. It seems like Markov ...
0
votes
1answer
37 views

sigma algebra of a stopping time

Let $N$ be a stopping time. i.e $\{N=n\} \in \mathbb{f}_n \forall n$. $\mathbb{f}_n$ is the filtration. $\mathbb{f}_N=\{A\in \mathbb{f}, A\cap \{N=n\}\in \mathbb{f}_n \forall n\}$ is the sigma ...
2
votes
0answers
24 views

A functional of a Lévy process

Does anyone know if there are any papers/results on functionals of the type : $$\int_0^tp(X_s)ds$$ where $X$ is a Lévy process and $p$ is a polynomial. For example, is the distribution of such an ...
2
votes
0answers
19 views

Stock Price Dynamics correlated with Bond market returns

I am currently working on to derive the following form of the stock price dynamics: $$dS_t = S_t[(r_t + \psi\sigma_S)dt + \rho \sigma_S dz_{1t} + \sqrt{1-\rho^2}\sigma_S dz_{2t}$$ where the ...
0
votes
1answer
26 views

Adapted and progressive processes

Could you please help me proving rigorously the following fact from Mayer's book: (a) if $X_t$ is a process adapted with respect to filtration $\{\mathcal{F}_t\}_{t\ge 0}$ and for every ...
2
votes
0answers
31 views

Stochastic process on compact spaces

I just heard some strange reasoning that I would like to understand with your help, let me describe the situation (unfortunately, I hesitated to ask the lecturer about it, because I apparently lacked ...
0
votes
0answers
25 views

Existence of compensator process under the assumption of local integrability and finite variation

I am reading a proof regarding existence of compensators under the assumption of local integrability in which I don't quite understand: Definition: The compensator of a cadlag adapted process $X$ ...
0
votes
0answers
10 views

Different definitions of local p integrability for local martingales

When talking about cadlag (but not continuous) martingales and local martingales in the context of stochastic integration one can come across different definitions depending on the author. These are: ...